# Linear Algebra

Linear Algebra

## Definiteness

A square matrix ${\displaystyle A}$ is positive definite if for all vectors ${\displaystyle x}$, ${\displaystyle x^{T}Ax>0}$.
If the inequality is weak (${\displaystyle \geq }$) then the matrix is positive semi-definite.

### Properties

• If the determinant of every upper-left submatrix is positive then the matrix is positive-definite.
• If ${\displaystyle A}$ is PSD then ${\displaystyle A^{T}}$ is PSD.
• The sum of PSD matrices is PSD.

### Examples

Examples of PSD matrices:

• The identity matrix is PSD
• ${\displaystyle xx^{T}}$ is PSD for any vector ${\displaystyle x}$